Pseudodual grids and extensions of generalized quadrangles. (Russian, English) Zbl 1014.05014

Sib. Mat. Zh. 42, No. 5, 1117-1124 (2001); translation in Sib. Math. J. 42, No. 5, 936-941 (2001).
Let \(\text{GQ}(s,t)\) be the class of all generalized quadrangles of order \((s,t)\) (i.e. a geometry consisting of points and lines such that each line comprises \(s+1\) points, each point lies on \(t+1\) lines, and for every point \(a\) not lying on a line \(L\) there exists a unique line passing through \(a\) and intersecting \(L\)). The subgraph \([a]\cap [b]\) is called a \(\mu\)-subgraph if the vertices \(a\) and \(b\) are at distance 2. Let \(\Gamma\) be an amply regular, locally \(\text{GQ}(s,t)\) graph, \(t>1\). If the \(\mu\)-subgraphs of \(\Gamma\) are pseudodual grids then it is proved that either \(s=t=2\) and \(\Gamma\) is a Taylor graph (a 2-antipodal cover of a clique) or \(\Gamma\) is the only strongly regular, locally \(\text{GQ}(2,4)\) graph with parameters \((64,27,10,12)\) (i.e. contains 64 vertices, is regular of valency 27, each of its edges lies in 10 triangles, and \([a]\cap [b]\) contains 12 vertices for every two vertices \(a\) and \(b\) that are at distance 2 in it).


05B25 Combinatorial aspects of finite geometries
51E12 Generalized quadrangles and generalized polygons in finite geometry
05E30 Association schemes, strongly regular graphs
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